I assume that the big, big dot in your post is the multiple sign.
You can use both rules (i.e, Chain Rule, and Product Rule) in this problem. There's no limit of the number of the rules you can use. So, just use it where you think is appropriated.
f(x) = (6 - x2) e2x
f'(x) = (6 - x2)' e2x + (6 - x2) (e2x)' = ...
You have taken the derivative of e2x with respect to x incorrectly.
We should use the Chain Rule there. By letting u = 2x, we have:
[tex]\frac{d}{dx} e ^ {2x} = \frac{d (e ^ u)}{du} \times \frac{du}{dx} = \frac{d(e ^ u)}{du} \times \frac{d(2x)}{dx} = ...[/tex]
So, can you see where your mistake is?
Can you go from here? :)